59.1.165 problem 167

Internal problem ID [9337]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 167
Date solved : Monday, January 27, 2025 at 06:01:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 43

dsolve(x^2*(1-2*x)*diff(y(x),x$2)+x*(8-9*x)*diff(y(x),x)+(6-3*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {48 c_{1} \left (-\frac {1}{2}+x \right )^{4} \left (x +\frac {4}{3}\right ) \sqrt {-1+2 x}+231 c_{2} \left (x^{3}-\frac {6}{7} x^{2}+\frac {2}{7} x -\frac {8}{231}\right )}{x^{6}} \]

✓ Solution by Mathematica

Time used: 0.512 (sec). Leaf size: 130

DSolve[x^2*(1-2*x)*D[y[x],{x,2}]+x*(8-9*x)*D[y[x],x]+(6-3*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} (3 x+4) \exp \left (\int _1^x-\frac {3 K[1]+4}{2 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\frac {8-9 K[2]}{K[2]-2 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {9 \exp \left (-2 \int _1^{K[3]}-\frac {3 K[1]+4}{2 K[1]-4 K[1]^2}dK[1]\right )}{(3 K[3]+4)^2}dK[3]+c_1\right ) \]